A bag contains four balls
of different colors (A of Red color, B Blue, C Green and D yellow). Select two
balls randomly one at a time and do not replace back to the bag after selecting. Calculate the total
number of combination of possible outcomes containing two balls selected without replacement.
Calculating the total number of outcomes using some objects selected randomly one at a time without replacement from the
total number of objects is important to calculate the probability of events. Because,
in sampling some sampling units are selected without replacement from the
population. Also, important is a mathematical concept of the ‘Combination’ used
in identifying the total number of possible outcomes given the condition and
calculating the probability of events. I show the combination of favorable outcomes
using the tree diagram 1.
Diagram 1: Combination of Outcomes in Selecting Two
Out of Four Balls Without Replacement
There will be two stages required
to select or arrange two out of four balls. At the first stage or draw, there
are four possibilities of randomly selecting the first ball. The first ball
could be one of red (A), blue (B), green (C) or yellow (D) balls shown by boxes
of those letters and colors in Diagram 1. Once the first ball, red (A), is
drawn and not replaced back, there will be three balls left in the bag, B, C
and D for the second stage or draw. If the second ball drawn is B, the first
outcome constituting two balls is AB which is numbered 1 on the right most side
of the diagram 1.
If the first ball selected is B and
not replaced back, there will be three balls left in the bag, A, C and D. At
the second stage or draw, there are three possibilities, one of B, C or D balls
for second stage or draw. If the second ball drawn is A, the outcome
constituting two balls is BA which is also numbered 1 on the right most side of
the diagram 1. Because they are same outcome if the position of the ball whether
the first or second is not meaningful. Thus, there are six combinations each with
two balls selected randomly without replacement from four balls.
Now, I will let you brainstorm how many will be the outcomes without replacement if there are six balls out of which two balls are drawn one at a time
without replacement? Could you use a diagram to show all possible outcomes? Perhaps,
difficult to show. Although the
diagrammatic logic is important to understand, as the number of balls or object
increases, the diagram becomes complicated and it will be less possible to show
all outcomes in the diagram. One may need to use the formula. Could you calculate
using a formula? If yes, that is fine. If not, wait for my other statistical notes.
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